Direct Simulation of the Motion of Particles in Flowing Liquids

Project Summary

The grand challenge proposed is to develop highly efficient methods for
computing the three-dimensional motions of large numbers of particles in
solid-liquid flows, under the action of the hydrodynamic forces and torques
exerted by the suspending fluid, and to use these methods to elucidate the
fundamental dynamics of particulate flows and solve problems of engineering
interest. The goal is to develop high-performance, state-of-the-art software
packages called particle movers, capable of simulating the motion of
thousands of particles in 2-D and hundreds in 3-D domains, in both Newtonian
fluids, governed by the Navier-Stokes equations, and in several popular models
of viscoelastic fluids. Such simulations will be extremely
computationally intensive. It is therefore imperative to develop the most
efficient possible computational schemes, and to implement them on parallel
machines, using state-of-the-art parallel algorithms.

To meet this challenge, we propose to develop two different computational
schemes for simulating solid-liquid flows on parallel computers. The first is a
generalization of the standard Galerkin finite element method in which both the
fluid and particle equations of motion are incorporated into a single
variational equation, containing both the fluid and particle velocities as
primitive unknowns. The hydrodynamic forces and torques on the particles are
eliminated in the formulation, so need not be computed as separate quantities.
The computation is performed on an unstructured body-fitted grid, and an
arbitrary Lagrangian-Eulerian moving mesh technique has been adopted to deal
with the motion of the particles.

In the second approach, an embedding method, the fluid flow is
computed as if the space occupied by the particles were filled with fluid. The
no-slip boundary condition on the particle boundaries is enforced as a
constraint using Lagrange multipliers. This allows a fixed grid to be
used, eliminating the need for remeshing, a definite advantage in parallel
implementations.

Both approaches have been initiated by us, for quite different kinds of
applications. At present, one scheme does not fit all applications. Perhaps
ultimately, a ``best'' universal scheme for moving particles may evolve, but it
is not presently prudent to make a bet.

A crucial computational issue to be addressed is the efficient solution of the
various algebraic systems which arise in the schemes. These systems can be
extremely large for 3-D problems, and their solution can consume up to 95 of
the CPU time of the entire simulation. It is therefore imperative to use
efficient iterative solution methods, with matrix-free preconditioners, and to
implement them on parallel architectures.

We plan to develop a library of parallel numerical algorithms to solve these
systems. This parallel library will consist of algorithms for solving nonlinear
algebraic equations using variants of Newton's method, preconditioned iterative
solvers for sparse symmetric indefinite and nonsymmetric linear systems, and
rapid elliptic and Stokes solvers on uniform grids. This library will be used
for rapid prototyping of simulation codes for the application problems referred
to above.

The library will be augmented with a collection of kernels to allow it to be
efficiently portable across either the massive parallelism of the Cray T3-D or
its successors, or cluster-based parallelism such as that of several
interconnected SGI Power-Challenge workstations. Both architectures exhibit
two-level parallelism that is ideally suited for schemes such as the embedding
method on a fixed grid.

The new schemes will be used to study the microstructural (pair interaction)
effects which produce clusters and anisotropic structures in particulate flows,
to produce statistical analyses of particulate flows (mean values, fluctuation
levels and spectral properties), to derive engineering correlations of the kind
usually obtained from experiments, and to provide clues and closure data for
the development of two-phase flow models and a standard against which to judge
the performance of such models. They will also be used to solve practical
problems of industrial interest such as sedimentation, fluidization and slurry
transport of solid particles in Newtonian and viscoelastic fluids.

Finally, the results of all numerical simulations will be compared with
experiments from the literature or experiments to be done in the Minnesota
laboratory, or with field data from industry---especially from our industrial
sponsors. The project will therefore advance the science of solid-liquid flow
using all the available tools: theory, experiments, and numerical simulation.